A Dunkl–Williams type inequality for absolute value operators

Dunkl and Williams showed that for any nonzero elements x,yx,y in a normed linear space XXxx-yy⩽4x-yx+y.Pečarić and Rajić gave a refinement and, moreover, a generalization to operators A,B∈B(H)A,B∈B(H) such that |A|,|B||A|,|B| are invertible as follows:|A|A|-1-B|B|-1|2⩽|A|-1(p|A-B|2+q(|A|-|B|)2)|A|-1where p,q>1p,q>1 with 1p+1q=1.In this note, we shall investigate the inequality and also equality conditions under the assumption that the existence of |A|-1|A|-1 and |B|-1|B|-1 is not required. Moreover, we give a refinement of the equality conditions.